(1) Property tax considers both land and land improvements, e.g. a house or factory. Taxing the house disincentivizes construction and investment on land, which is exactly the opposite of what we want. The land value tax only attempts to capture rent, which is an intrinsic value associated with the productivity of the location compared to alternatives. Actual studies of this policy have been done and LVT encourages construction when used to displace property taxes.

(2) A tax on the market value of the land (or property) can never capture the full ground rent. This is because the market value of land derives from the rent you could collect from it (or the work you could do on it yourself). The market value of the land will decrease, adjusting for the lost tax revenue, and that in turn changes the amount of tax that will be collected...

Let's call "Rev" the annual tax revenue, "Rate" the annual tax rate on land value, "Rent" the true ground rent, and "t" the number of years someone considers in their assessment of market value of real estate.

Rev = (market_value)*Rate
Rev = (Rent*t - Rev*t)*Rate
Rev = Rent * t*Rate/(1 + t*Rate)

No matter what combination of t and Rate you choose, the tax revenue will always be smaller than ground rent. Moreover, the higher you try to set the tax rate, the more unstable the assessment will get, because you're trying to charge large multiples of tiny appraisal values. Instead you have to assess the ground rent directly.